We first rewrite the two expressions in the given equation to the same base

3x2 / 34x = 3x2-4x

and

1/81 = 1 / 34 = 3-4

We now rewrite the equation as follows

3x2-4x = 3-4

Which leads to the following algebraic equation

x2 - 4x = -4

x2-4x + 4 = 0

factor and solve

(x - 2)2 = 0

x = 2

What are the solutions of x(x - 2) = 5?

Solution

Expand and write in standard form

x2 - 2x - 5 = 0

Use quadratic formulas

a
x = [ -(-2) + or - sqrt( (-2)2 - 4(1)(-5)) ] / 2

= [ 2 + or - sqrt(24) ] / 2

= [ 2 + or - 2 sqrt(6) ] / 2

two solutions: 1 + sqrt(6) and 1 - sqrt(6)

The formula for the nth term, an, of an arirhmetic progression is given by an = a1 + (n - 1)d, where a1 represents the first term of the progession and d represents its common difference. What is the value of the 20th term of the arithmetic progression 4, 7, 10,...?

Solution

The common difference of the given progression is

7 - 4 = 3 (or 10 - 7 = 3)

The first term is

a1 = 4

The value of the 20th term is given by

a20 = a1 + (n - 1)d = 4 + (20 - 1)3 = 4 + 3 × 19 = 4 + 57 = 61

In a a 16-by-12 rectangle, what is the perimeter of the triangle formed by two sides of the rectangle and the diagonal?

Solution

Let x be the length of the diagonal and use Pythagora's theorem to find it.

x2 = 162 + 122 = 256 + 144 = 400

x = sqrt(400) = 20

The three sides of the triangle are the length and width of the rectangle and the diagonal.

Which of these interval represents all the real values that are the range of y = 1 / (4 - x2)

Solution

y = 1 / (4 - x2) is a even function since

y(-x) = 1 / (4 - (-x)2) = 1 / (4 - x2)

Because it is an even function and therefore its graph is symmetric with respect to the y axis, we shall study its graph for x ≥ 0.
Since the denominator of this rational function is zero at x = + or -2, it has 2 vertical asymptotes: x = 2 and x = - 2.
It also has a horizontal asymptote given by y = 0 because the degree of the denominator is greater than the degree of the numerator.